I would like to know how the projection functor $K(A) \to D(A)$ induces an equivelance $A-\textrm{cofib} \sim D(A)$, where $A-\textrm{cofib}$ is the full subcategory of $K(A)$ on the cofibrant objects.

The cannonical projection from $K(A) \to D(A)$ admits a left adjoint functor ${\bf p}$ sending a dg $A$-module to a cofibrant dg $A$-module $_{\bf p}M$ quasi-isomorphic to $M$. What's the left adjoint functor ${\bf p}$ or how to understand the left adjoint functor ${\bf p}$?

$\begingroup$The notation $K(A)$ for the homotopy category is unusual, and there is no reasonable functor from the homotopy category to $D(A)$. By $K(A)$ do you mean the category of cofibrant objects?$\endgroup$
– Dmitry VaintrobDec 17 '16 at 15:21

$\begingroup$I mean that $K(A)$ is a homotopy category, see en.wikipedia.org/wiki/Derived_category $A$-cofib is the full subcategory of $K(A)$ consisting of cofibrant objects.$\endgroup$
– bingDec 18 '16 at 12:07